From Properties to a Rigidity Conjecture on Finsler via a Detecting Algorithm

Alexandru Kristály and Ágoston Róth

Abstract. The aim of this paper is to consider Busemann-type inequalities on Finsler manifolds. We actually formulate a rigidity conjecture: any Finsler which is a Busemann NPC is Berwaldian. This statement is supported by some theoretical results and numerical examples. The presented examples are ob- tained by using evolutionary techniques (genetic algorithms) which can be used for detecting on a large class of not necessarily reversible Finsler manifolds.

2000 Mathematics Subject Classi…cation: 58B20, 53C60, 53C70, 32Q05. Keywords: Busemann NPC space, Finsler manifold, Berwald space, numerical detection of geodesics via genetic algorithms.

1 Introduction

In the forties, Busemann developed a synthetic geometry of G-spaces, see [3]. These spaces possess the essential geometric properties of Finsler manifolds but they have no necessarily a di¤erential structure. Nowadays, Busemann’scurvature is referred to non-positively curved (quasi-)metric spaces, called Busemann NPC spaces; this notion of non-positive curvature requires that in small geodesic triangles the length of a side is at least the twice of the geodesic distance of the mid-points of the other two sides, see [3, p. 237]. Note that on Riemannian manifolds the non-positivity of the sectional curvature characterizes Busemann’snon-positivity curvature. In the …fties, Aleksandrov introduced independently another notion of curvature on metric spaces, a stronger one than that of Busemann, see Jost [5, Section 2.3]. Nevertheless, on Riemannian manifolds the Aleksandrov curvature condition holds if and only if the sectional curvature is non-positive, see Bridson-Hae‡iger[2, Theorem 1A.6]. In the Finsler case the picture is quite rigid: if on a reversible Finsler manifold (M;F ) the Aleksandrov curvature condition holds (on the induced metric space by (M;F )) then (M;F ) should be Riemannian, see [2, Proposition 1.14]. Having these facts in our mind, a natural question arises: What about Finsler manifolds that are Busemann NPC spaces? Clearly, we are thinking …rst to those Finsler manifolds which have non-positive ‡ag curvature. However, within this class, we easily …nd examples which are not Busemann NPC spaces. For instance, the Hilbert metric of the interior of a simple, closed and strictly convex c in the Euclidean plane de…nes a Finsler metric with constant ‡ag curvature 1, but Busemann’s curvature condition holds if and only if c is an ellipse, see Kelly-Straus [6]. In the latter case, the Hilbert metric becomes a Riemannian one. The above example suggests we face again a rigidity result as in the Aleksandrov case. However, a recent result of Kozma-Kristály-Varga [8] entitle us to believe that the appropriate class of Finsler manifolds is the Berwald one, a larger class than Riemannian manifolds. Namely, we formulate the following

Conjecture. Let (M;F ) be a (not necessarily reversible) Finsler manifold such that (M; dF ) is a Busemann NPC space. Then (M;F ) is a Berwald space.

The aim of the present paper is to recall some theoretical developments and certain numerical examples in order to support the above Conjecture. In addition, we propose a new numerical approach to detect minimal geodesics on Finsler manifolds by using evolutionary techniques. Note that the presented numerical approach can be e¢ ciently used for verifying di¤erent metric relations on a large class of Finsler manifolds exploiting the basic property of Finslerian-geodesics that they locally minimize the integral length. The paper is organized as follows. In the next section we recall the notion of Busemann NPC (quasi-)metric spaces and basic elements of Finsler manifolds. Section 3 deals with the Conjecture from both theoretical and numerical point of view. Section 4 is independent from the previous ones and describes the detection of geodesics on Finsler manifolds by using genetic algorithms.

1 2 Preliminaries 2.1 Busemann NPC spaces Let (M; d) be a quasi-metric space (i.e., d is not necessarily symmetric). A continuous curve c : [0; r] M with c (0) = p; c (r) = q is a shortest geodesic, if l (c) = d (p; q), where l (c) is the generalized length of !c and it is de…ned by n l (c) = sup d (c (ti 1) ; c (ti)) : 0 = t0 < t1 < ::: < tn = r; n N : 2 (i=1 ) X We say that (M; d) is a locally geodesic (length) space if for every point p M there is a neighborhood 2 Np M such that for every two points x; y Np there exists a shortest geodesic joining them. In the sequel, we always assume that the shortest geodesics2 are parametrized proportionally to the arc length, i.e., l c = t l (c) ; t [0; r] : j[0;t]  8 2 De…nition  1 (Busemann NPC space) A locally geodesic space (M; d) is said to be a Busemann non-positive curvature space (shortly, Busemann NPC space), if for every p M there exists a neighborhood Np M 2  such that for any two shortest geodesics c1; c2 : [0; 1] M with c1 (0) = c2 (0) Np and with endpoints ! 2 c1 (1) ; c2 (1) Np we have 2 1 1 2d c1 ; c1 d(c1 (1) ; c2 (1)): (1) 2 2       2.2 Finsler manifolds

Let M be a connected m-dimensional C1 manifold and let TM = p M TpM be its . If the 2 continuous function F : TM [0; ) satis…es the conditions that it is C1 on TM 0 ; F (p;ty) = tF (p; y) for ! 1 S nf g 1 2 all t 0 and y TpM; i.e., F is positively homogeneous of degree one; and the matrix gij(y) := F (y)  2 2 yiyj is positive de…nite for all y TM 0 ; then we say that (M;F ) is a Finsler manifold. If F is absolutely   homogeneous, then (M;F ) is2 said ton be f greversible. Let TM be the pull-back of the tangent bundle TM by  : TM 0 M: Unlike the Levi-Civita connection in Riemann geometry, there is no unique natural connectionn in f g the ! Finsler case. Among these k connections on TM; we choose the Chern connection whose coe¢ cients are denoted by ij; see Bao-Chern- Shen [1, p. 38]. This connection induces the curvature tensor, denoted by R; see [1, Chapter 3]. The Chern connection de…nes the covariant derivative DVU of a vector …eld U in the direction V TpM: Since, in i 2 general, the Chern connection coe¢ cients jk in natural coordinates have a directional dependence, we must say explicitly that DVU is de…ned with a …xed reference vector. In particular, let c : [0; r] M be a smooth k i ! curve with velocity …eld T = T(t) = c_(t) = T k=1;m : Suppose that U = U i=1;m and W are vector …elds de…ned along c: We de…ne D U with reference vector W as T   i dU j k i @ DTU = + U T (jk)(c;W) i ; dt @p c(t)   j

@ where i is a basis of Tc(t)M: A C1 curve c : [0; r] M; with velocity T = c_ is a (Finslerian) @p c(t) j i=1;m ! geodesicn if o T D = 0 with reference vector T. (2) T F (T)   If the Finslerian speed of the geodesic c is constant, then (2) becomes

d2ci dcj dck + i = 0; i = 1; :::; m = dim M: (3) dt2 dt dt jk (c;T)  In the sequel, we assume that the geodesics have constant Finslerian speed.

2 A Finsler manifold (M;F ) is said to be forward (resp. backward) geodesically complete if every geodesic c : [0; 1] M can be extended to a geodesic de…ned on [0; ) (resp. ( ; 1]). ! 1 1 Let c : [0; r] M be a piecewise C1 curve. Its integral length is de…ned as ! r LF (c) = F (c (t) ; c_ (t)) dt: Z0

For p; q M, denote by (p; q) the set of all piecewise C1 c : [0; r] M such that c (0) = p and 2 [0;r] ! c (r) = q. De…ne the map dF : M M [0; ) by  ! 1

dF (p; q) = inf LF (c) : (4) c (p;q) 2 [0;r]

Of course, we have dF (p; q) 0; where equality holds if and only if p = q; and the triangle inequality  holds, i.e., dF (p0; p2) dF (p0; p1) + dF (p1; p2) for every p0; p1; p2 M: In general, since F is only a  2 positive , dF (p; q) = dF (q; p); thus, (M; dF ) is only a quasi-metric space. If (M; g) is a 6 , we will use the notation dg instead of dF which becomes a usual metric function. Busemann and Mayer [4, Theorem 2, p. 186] proved that the generalized length l(c) and the integral length LF (c) of any (piecewise) C1 curve coincide for Finsler manifolds. Let (p; y) TM 0 and let V be a section of the pulled-back bundle TM. Then, 2 n f g

g(p;y) (R (V; y) y; V) K (y; V) = 2 (5) g (y; y) g (V; V) g (y; V) (p;y) (p;y) (p;y) is the ‡ag curvature with ‡ag y and transverse edge V. Here,  

i j 1 2 i j g(p;y) := gij(p;y)dp dp := F dp dp ; p M; y TpM; (6) 2 i j 2 2  y y

is the Riemannian metric on the pulled-back bundle TM. In particular, when the Finsler structure F arises 1 2 from a Riemannian metric g (i.e., the fundamental tensor gij = F does not depend on the direction y), 2 yiyj the ‡ag curvature coincides with the usual sectional curvature.   If K (V; W) 0 for every 0 = V; W TpM; and p M, with V and W not collinear, we say that the ‡ag curvature of (M;F ) is non-positive6 . 2 2 k A Finsler manifold is of Berwald type if the Chern connection coe¢ cients ij in natural coordinates depend only on the base point. Special Berwald spaces are the (locally) Minkowski spaces and the Riemannian manifolds.

3 Busemann NPC spaces versus Finsler manifolds: the Conjecture 3.1 Theoretical results

Let (M; g) be a Riemannian manifold and dg the metric function induced by g. In the …fties, Busemann proved the following result.

Theorem 2 ([3, Theorem (41.6)]) Let (M; g) be a Riemannian manifold. The metric space (M; dg) is a Busemann NPC space if and only if the sectional curvature of (M; g) is non-positive.

Due to Theorem 2, Shen [12] formulated an open problem on the existence of a curvature notion on a B Finsler manifold (M;F ) with the property that 0 if and only if (M; dF ) is a Busemann NPC space. For Finsler manifolds, as far as we know, the unique resultB  concerning Busemann-type inequalities is due to Kozma, Kristály and Varga (cf. [8] and [7]).

Theorem 3 Let (M;F ) be a (not necessarily reversible) Berwald space with non-positive ‡ag curvature. Then (M; dF ) is a Busemann NPC space.

3 Remark 4 Beside Riemannian manifolds, Theorem 3 easily follows for (not necessarily reversible) Minkowski spaces. Indeed, since M is a vector space and F is a Minkowski inducing a Finsler structure on M by translation, its ‡ag curvature is identically zero, the geodesics are straight lines, and in the Busemann-type inequality (see De…nition 1) we always have equality.

Note that there are Berwald spaces with non-positive ‡agcurvature which are neither Riemannian manifolds nor Minkowski spaces. In the sequel, we give such an example. Let (N; h) be an arbitrarily closed hyperbolic Riemannian manifold of dimension at least 2; and " > 0. Let us de…ne the Finsler metric F" : T (R N) [0; ) by  ! 1

2 2 4 F"(t; p; ; w) = hp (w; w) +  + " hp (w; w) +  ; r q where (t; p) R N and (; w) T(t;p)(R N): Shen [9] pointed out that the pair (R N;F") is a reversible Berwald space2 with non-positive2 ‡ag curvature.  The proof of Theorem 3 exploits the fact that the Finsler manifold we are dealing with is actually a Berwald space: (a) the reverse of a given geodesic segment is also a geodesic (which is not true for every non-reversible Finsler manifold); (b) the reference vectors are not relevant since the Chern connection coe¢ cients do not depend on the direction. We conclude this subsection with the following simple observation which is based on the Cartan-Hadamard theorem [1, p. 238], see also [5, Corollary 2.2.4].

Proposition 5 Let (M;F ) be a (not necessarily reversible) forward or backward geodesically complete, simply connected Finsler manifold such that (M; dF ) is a Busemann NPC space. Then (M; dF ) is a global Busemann NPC space, i.e., inequality (1) holds for any pair of geodesics.

3.2 Supporting the Conjecture: Examples In the sequel we provide some relevant examples which support the Conjecture beside the theoretical results presented in the previous subsection.

1. Hilbert geometry. We consider the Hilbert metric of the interior of a simple, closed and strictly convex 2 curve c in the Euclidean plane. In order to describe this metric, let Mc R be the region de…ned by the  interior of the curve c and …x p1; p2 Mc. Assume …rst that p1 = p2. Since c is a convex curve, the straight 2 6 line passing to the points p1; p2 intersects the curve c in two points; denote them by u1; u2 c: Then, there 2 are 1; 2 (0; 1) such that pi = iu1 + (1 i) u2 (i = 1; 2). The Hilbert distance between p1 and p2 is 2

1 1 2 dH (p1; p2) = log : 1     2 1 

We complete this de…nition by dH (p; p) = 0 for every p Mc. One can easily prove that (Mc; dH ) is a metric space and de…nes a reversible, projective Finsler metric 2

dH (p; p + ty) FH (p; y) = lim t 0+ t ! with constant ‡ag curvature 1. However, due to Kelly-Straus, we have 2 Proposition 6 (see [6]) The metric space (Mc; dH ) is a Busemann NPC space if and only if the curve c R is an ellipse. 

Consequently, (Mc; dH ) is a Busemann NPC metric space if and only if (Mc;FH ) is a Riemannian manifold.

4 2. A projectively ‡at Finsler metric with negative ‡ag curvature. Fix " [ 1; 1). For any p m m m m 2 2 B (0;1) = p R : p < 1 and y TpB (0;1) = R we de…ne f 2 j j g 2

2 2 2 2 1 y ( p y p; y ) + p; y F (p; y) = j j j j j j h i h i (7) " 2 8q 1 p 2 < j j 2 2 2 2 " :y "2( p y p; y ) + "2 p; y j j j j j j h i h i : q 1 "2 p 2 9 j j = m The pair (B (0;1) ;F") is a forward geodesically complete Finsler manifold for any; " ( 1; 1). When " = 1, m m 2 F 1 is the well-known Klein/Hilbert metric on B (0;1) and B (0;1) ; dF 1 is a global Busemann NPC space, see also the previous example for m = 2. Note that F" (" [ 1; 1)) is projectively ‡at with constant m2  ‡ag curvature 1; see Shen [11, p. 1722]. When " ( 1; 1), (B (0;1) ;F") is not Berwaldian. On account m 2 m of Proposition 5, if (B (0;1) ; dF" ) is a Busemann NPC space, the quasi-metric space (B (0;1) ; dF" ) is a global Busemann NPC space. However, Figure 1 (a) shows that one can easily …nd (large) geodesic triangles where Busemann-type inequality fails. This fact does not imply that Busemann-type inequality cannot hold in particular, see Figure 1 (b).

Figure 1: Testing the Busemann-type inequality in case of the Finsler metric (7) for " = 0:1. In case of the geo- desic triangle (a) the Busemann-type inequality does not hold. The nodes of the triangle are p0 (0:0; 0:0; 0:9), p1 (0:5; 0:5; 0:5) and p2 ( 0:5; 0:5; 0:5). The lengths of the sides and of the median are dF (p0; p1) = " 1:14416728, dF" (p0; p2) = 0:679288454, dF" (p1; p2) = 1:23015501 and dF" (m1; m2) = 0:921800308, respec- tively. However, in case of the geodesic triangle (b) the Busemann-type inequality holds. The nodes of this tri-

angle were obtained from triangle (a) by interchanging the nodes p0 and p1. In this case we have dF" (p0; p1) =

1:28109422, dF" (p0; p2) = 1:23015501, dF" (p1; p2) = 0:679288454 and dF" (m1; m2) = 0:312663329.

5 3. Finslerian-Poincaré disc. Let us consider the disc

2 2 2 M = (x1; x2) R : x1 + x2 < 4 : 2

Introducing the polar coordinates (r; ) on M , i.e., x1 = r cos , x2 = r sin , we de…ne the non-reversible Finsler metric on M by 1 pr V 2 2 2 F ((r; ) ; ) = r2 p + r q + r4 ; 1 4 1 16 p where @ @ V = p + q T M: @r @ 2 (r;) The pair (M;F ) is the so-called Finslerian-Poincaré disc. Within the classi…cation of Finsler manifolds, (M;F ) is a Randers space, see [1, Section 12.6], which is a forward (but not backward) geodesically complete manifold having constant negative ‡ag curvature 1=4. Due to Szabó’srigidity theorem about Berwald surfaces (see for instance [1, Theorem 10.6.2]), (M;F ) is not a Berwald space. If (M; dF ) is a Busemann NPC space, due to Proposition 5, the quasi-metric space (M; dF ) is a global Busemann NPC space. However, Figure 2 shows that one can …nd geodesic triangles where Busemann-type inequalities do not hold.

Figure 2: Two geodesic triangles in Finslerian-Poincaré disc from many others for which the Busemann-type inequality fails. (a) The nodes of the geodesic triangle are p0 (1; 1), p1 ( 1; 1) and p2 ( 1; 1). The lengths of the sides and of the median are dF (p0; p1) = 3:5254963, dF (p0; p2) = 2:88728397, dF (p1; p2) = 2:88728399 and dF (m1; m2) = 1:71860536, respectively. (b) Another geodesic triangle the nodes of which are p0 (1; 0), p1 (1; 1) and p2 (1; 1). In this case we have dF (p0; p1) = 2:07878347, dF (p0; p2) = 2:07878333, dF (p1; p2) = 2:88728393 and dF (m1; m2) = 1:70407783.

6 4. A non-projectively ‡atFinsler metric with zero ‡agcurvature. Let m 2 and consider the cylinder  2 m 2 2 2 M = p = ((x1; x2) ; x3) R R : x1 + x2 < 1 : 2   m For any y = ((y1; y2) ; y3) TpM = R we de…ne 2 e

2 2 2 2 e ( x2y1 + x1y2) + y (1 x1 x2) ( x2y1 + x1y2) F (p; y) = j j : (8) q 1 x2 x2 1 2 The pair (M;F ) is an m-dimensional Finsler manifold with constant ‡ag curvature 0, see Shen [10, Theorem 1.1] and [9]. Moreover, (M;F ) is not Berwaldian and it is not forward geodesically complete. For m = 2 and m = 3, we present some constellations of small geodesic triangles where Busemann-type inequality does not hold, see on Figure 3 (a) and (b), respectively. Due to the lack of geodesically completeness, we cannot apply Proposition 5. Therefore, we have to consider "small" geodesic triangles which are inside of a normal neighborhood of some point p M (see the set Np from De…nition 1). At the moment we cannot estimate the 2 size of the normal neighborhood Np for a …xed point p M. 2

Figure 3: Two geodesic triangles are presented from many others for which the Busemann-type inequality fails in the 2- and 3-dimensional cases of the Finsler metric (8). (a) The nodes of the geodesic triangle are p0 ( 0:25; 0:5), p1 (0:875; 0) and p2 ( 0:4975; 0:5). The lengths of the sides and of the median are dF (p0; p1) = 0:845022533, dF (p0; p2) = 1:2647537, dF (p1; p2) = 0:999184207 and dF (m1; m2) = 0:629171204, respec- tively. (b) Another geodesic triangle the nodes of which are p0 ( 0:25; 0:5; 1:0), p1 ( 0:4975; 0:5; 1:0) and p2 (0:995; 0:0; 0:5). In this case we have dF (p0; p1) = 2:28325263, dF (p0; p2) = 1:59552717, dF (p1; p2) = 1:63538395 and dF (m1; m2) = 1:03707673.

7 4 Finding Finslerian-geodesics via evolutionary techniques

Evolutionary techniques are useful and e¢ cient in case of such optimization problems when the search space is large, complex or traditional search and numerical methods fail. They are based on principles of the evolution via natural selection, employing a population of individuals that undergo selection in the presence of variation- inducing operators such as mutation and recombination. Consider the m-dimensional (m 2) Finsler manifold (M;F ) and the set of curves  k 2 (p; q) = c C ([0; 1] ;M): c (0) = p; c (1) = q ; [0;1] 2 where the order k 2 of continuity is at least 1 and the points p; q M are preliminary …xed. At the moment, 2 the proposed algorithm provides results when M Rm. A genetic algorithm is presented as a computer simulation in which isolated populations of candidate solu- tions to the optimization problem

1 LF (c) = F (c (t) ; _c (t)) dt min ! 8 Z0 (9) > < c (p; q) 2 [0;1] > evolve towards better solutions. From:> Finsler geometrical point of view we exploit the fact that short geodesics minimize the integral length, see [1, Theorem 6.3.1]. Thus, in the sequel we implicitly assume that p and q are close enough to each other in the sense that they belong to a common normal neighborhood Np of a certain point p M. In fact2 the algorithm is a generational process (controlled by selection, recombination/crossovere and muta- tion) thate is repeated until a termination condition has been reached. Common terminating conditions are: a solution is found that satis…es minimum criteria, a …xed number of generations is reached, the …tness of the highest ranking solution is at a such a level that successive iterations no longer produce better results, etc. Before starting the genetic representation of the individuals that are involved in the search processes, we recall the de…nition and some properties of B-spline curves that will be used in our further investigations.

k De…nition 7 (Normalized B-spline basis functions) The recursive function Nj (t) (k 1) given by the equations  1; t [t ; t ) N 1 (t) = j j+1 j 0; otherwise2  k t tj k 1 tj+k t k 1 Nj (t) = Nj (t) + Nj+1 (t) ; t R; tj+k 1 tj tj+k tj+1 2 is called normalized B-spline basis function of order k (degree k 1). The real numbers tj tj+1 are called 0 :  knot values or simply knots and 0 = 0 by de…nition. De…nition 8 (B-spline curves) The curve

n k k c (t) = djNj (t) ; t [tk 1; tn+1) (10) 2 j=0 X k is called B-spline curve of order 1 k n + 1 (degree k 1), where Nj (t) is the jth normalized B-spline basis function for the evaluation of which the knots

t0 t1 ::: tn+k    m are necessary. The points dj R (j = 0; 1; : : : ; n; m 2) are called control points (de Boor-points), while the polygon 2  n m D = [dj] 1;n+1 (R ) j=0 2 M

8 is called control polygon (de Boor-polygon). The arcs of this B-spline curve are called spans and the ith (i = k 1; k; : : : ; n) span can be expressed as i k k c (t) = djN (t) ; t [ti; ti+1) : (11) i j 2 j=i k+1 X k k The normalized B-spline basis functions Nj (t) are positive. Moreover, Nj (t) = 0 if t = [tj; tj+k) and the k 2 basis functions form a partition of the unity, i.e., j Nj (t) 1; t R. Thus, any point of the curve (10) is  8 2 k k located in the convex hull of k consecutive control points. Moreover, the consecutive spans ci 1 (t) and ci (t) k 2 P (i = k; : : : ; n) are C -continuous at the knot value t = ti provided that k 2 and the multiplicity of ti is one. The following subsections describe the genetic representations of individuals and search operators involved in this genetic algorithm which proved to be useful for the optimization problem (9).

4.1 Genetic representation Initial individuals of the population involved in the search process are randomly generated B-spline curves. Each individual is determined by an order k, a control polygon

n m Q = [d0; d1; :::; dn] = [dj] 1;n+1 (R ) j=0 2 M and a knot vector T = t0; t1; : : : ; tn+k : f g Thus, an individual can be denoted as the triplet (k; Q;T ). Knot values

t0 = t1 = ::: = tk 1 = 0 < tk < tk+1 < : : : < tn < tn+1 = tn+2 = ::: = tn+k = 1

and control points p; j = 0; 8 dj = > d (j; p; q; j) ; j 1; 2; : : : ; n 1 ; > 2 f g <> q; j = n > > ensure the required endpoint interpolation:> property, where the interior control points dj = d (j; p; q; j) (j = 1; 2; : : : ; n 1) are selected randomly within the m 1 dimensional Euclidean balls m 1 m 1 B (oj; j) = x R : x oj < j ; j = 1; 2; : : : ; n 1 2 j j which are the subsets of the hyperplanes that are perpendicular (in Euclidean sense) to the direction determined by the points p; q and the origin of which are determined as

j j oj = 1 p + q; j = 1; 2; : : : ; n 1: n n   Figure 4 illustrates these balls in 2- and 3-dimensional cases. Naturally, one must also ensure that the image of all randomly generated individuals (B-spline curves) are the part of the Finsler manifold (M;F ). The radius j m 1 (j = 1; 2; : : : ; n 1) of the Euclidean ball B (oj; j) is a user-de…ned and manifold dependent …xed number. Knot locations also e¤ect the shape of the B-spline curve, but in this case the interior knot values tk; tk+1; : : : ; tn will be uniformly chosen, i.e., j k + 1 t = ; j = k; k + 1; : : : ; n: j n k + 2

9 Figure 4: Generating an initial individual: (a) and (b) illustrate the domains of the possible interior control points in 2- and 3-dimensional cases.

The number n + 1 of control points, the order k and the knot vector T of individuals are the same for all individuals and they do not change during the evolutionary process. Thus, control points within a control polygon are the genes of an individual. A search process starts with a randomly generated initial population

P0 = (k; Q1;T ) ; (k; Q2;T ) ;:::; (k; QN ;T ) f g consisting of the individuals l l l (k; Ql;T ) = k; d0; d1;:::; dn ;T ; l = 1; 2;:::;N:

As the population evolves from the current  generation Pg (g 0) to the next generation Pg+1, the number N of individuals remains the same.  The …tness value of the individual (k; Ql;T ) ; l = 1; 2; :::; N is given by k eval [(k; Ql;T )] = LF c ; l where n  k l k cl (t) = djNj (t) ; t [tk 1; tn+1) 2 j=0 X is a B-spline curve of order k that consist of the spans i k l k c (t) = d N (t) ; t [ti; ti+1) ; i = k 1; k; : : : ; n; l;i j j 2 j=i k+1 X i.e., k eval [(k; Ql;T )] = LF cl n  k = LF c l;i i=k 1 X  n ti+1 = F ck (t) ; _ck (t) dt: l;i l;i i=k 1 Zti X  Individual (k; Qi;T ) is considered better than individual (k; Qj;T ) if

eval [(k; Qi;U)] > eval [(k; Qj;T )] :

10 4.2 Search operators In the sequel we give a short description of search operators involved in the proposed genetic algorithm.

4.2.1 Selection Tournament selection is used for determining the individuals of the next generations. This selection mechanism runs a "tournament" among a few individuals chosen randomly from the current population and selects the "winner" (i.e., the one with the best …tness) for recombination (or crossover). A number N of tournaments will result in N winner individuals that will be inserted into a mating pool. Individuals within this mating pool are used to generate new o¤springs.

4.2.2 Recombination

Let us consider a randomly coupled pair ((k; Qi;T ) ; (k; Qj;T )) (i = j, i; j 1; 2;:::;N ) of individuals (called 6 2 f g parents) from the mating pool. In order to control this process, a recombination probability prec [0; 1] can be used that determines how often will crossover be performed. Recombination is based on the following2 mating process: …rst we randomly select an interior point of the parents’control polygons, then the complementary parts of the parents’control points in arrays determined by the selected point are used to form the genes of the o¤springs in the following way:

i i i i i i i i j j Qi = d0; d1; :::; dl 1; dl; :::; dn Qi0 = d0; d1; :::; dl 1; dl ; :::; dn j j = 8 h i  j j j j  9 ) Q = d ; d ; :::; d ; d ; :::; dj > j j j i i j 0 1 l 1 l n => <> Q0 = d ; d ; :::; d ; d ; :::; d : j j 0 1 l 1 j l n h i > > h i The resulted o¤springs (called children), the parents; :> and the worst individual of the current population compete for survival. Naturally, one can also de…ne multiple point recombination operators.

4.2.3 Mutation Mutation is an operator used to maintain genetic diversity from the current generation of a population to the next one. This process can also be controlled by a mutation probability pmut (0; 1] that determines which genes will be altered. 2 Consider the randomly selected individual

l l l (k; Ql;T ) = k; d ; d ; :::; d ;T ; l 1; 2; :::; N ; 0 1 n 2 f g l    and let us suppose that control point dj, j 1; :::; n 1 has also been randomly selected for mutation. The selected control point lies in a hyperplane that2 f is perpendicular g (in Euclidean sense) to the direction of q p. Consider the m 1 dimensional Euclidean ball m 1 l m 1 l B dj; mut = x R : x dj < mut ; 2   in the aforementioned hyperplane, where mut > 0 is called maximal mutational radius, which is either a …xed or a generation-based descending number that speci…es the maximum deviation of the point being under mutation. l A possible mutation is done by perturbing control point dj such that its new position will be somewhere in m 1 l the Euclidean ball B dj; mut . Naturally, we need also to ensure that after the perturbation process the image of the new individual (B-spline curve) will fall still in the manifold (M;F ). Once again, the resulting o¤spring, the parent and the worst individual of the current population compete for survival. Figure 5 depicts the evolution of four isolated populations the best individuals of which converge to the sides and the median of a geodesic triangle in the Finslerian-Poincaré disc.

11 Figure 5: Evolution of the geodesic triangle in the Finslerian-Poincaré disc (see Figure 2). Three isolated populations of B-spline curves evolve to the expectant sides: one can follow up from (a) to (f) the evolving shape of the best individuals in consecutive generations of the corresponding populations. In case of the evolved geodesic triangle a new isolated population evolves the median (cf. images from (g) to (i)).

12 5 Final remarks

1. During the preparation of the manuscript we tested much more Finsler metrics (Berwald and non-Berwald structures) than those presented in the previous sections. We emphasize that the proposed numerical method can be e¢ ciently applied to check various (local) metric relations on Finsler manifolds not only for Busemann-type inequalities. 2. The time and memory complexities of the proposed genetic algorithm heavily depend on the number of individuals of each isolated populations that evolve towards di¤erent geodesic lines. The evaluation of the points and derivatives of the B-spline curves depends on the order of the curves and on the size of the control polygons. The used quadrature formula, that approximates the theoretical value of the integral length needed for …tness evaluation, also e¤ects the runtime of the algorithm. However, in case of the provided examples all geodesic triangles and their medians were obtained in reasonable time (i.e., approximately 40-120 seconds on a computer having an Intel Core 2 Duo 2.8 GHz processor).

Acknowledgement

Both authors were supported by Grant CNCSIS PNII-527/2007.

References

[1] D. Bao, S. S. Chern, Z. Shen, 2000. An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, 200, Springer Verlag.

[2] M. Bridston, A. Haefliger, 1999. Metric spaces of non-positive curvature, Springer–Verlag, Berlin. [3] H. Busemann, 1955. The Geometry of Geodesics, Academic Press. [4] H. Busemann, W. Mayer, 1941. On the foundations of the calculus of variations, Transactions of the American Mathematical Society, 49(2):173–198.

[5] J. Jost, 1997. Nonpositivity Curvature: Geometric and Analytic Aspects, Birkhäuser Verlag, Basel. [6] P. Kelly, E. Straus, 1958. Curvature in Hilbert geometry, Paci…c Journal of Mathematics, 8(1):119–125. [7] A. KristÁly, L. Kozma, 2006. Metric characterization of Berwald spaces of non-positive ‡ag curvature, Journal of Geometry and Physics, 56(8):1257–1270.

[8] A. KristÁly, Cs. Varga, L. Kozma, 2004. The dispersing of geodesics in Berwald spaces of non-positive ‡ag curvature, Houston Journal of Mathematics, 30(2):413–420.

[9] Z. Shen, 2005. Finsler manifolds with nonpositive ‡ag curvature and constant S-curvature, Mathematische Zeitschrift, 249(3):625–639.

[10] Z. Shen, 2003. Finsler metrics with K = 0 and S = 0, Canadian Journal of Mathematics, 55(1):112–132. [11] Z. Shen, 2003. Projectively ‡at Finsler metrics of constant ‡ag curvature, Transactions of the American Mathematical Society, 355(4):1713–1728.

[12] Z. Shen, January 7, 2009. Some open problems in Finsler geometry, http://www.math.iupui.edu/zshen/Research/papers/Problem.pdf.

Alexandru Kristály: Babe¸s-Bolyai University, Department of Economics, RO–400591 Cluj-Napoca Ágoston Róth: Babe¸s-Bolyai University, Chair of Numerical and Statistical Calculus, RO-400084 Cluj-Napoca

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